January 8th, 2017, 12:34 PM  #1 
Newbie Joined: Jan 2017 From: los angeles Posts: 2 Thanks: 0  optimization problem
Consider an optimization problem, minimize f(Ax)+x^T x the variable is nvector x. The matrix A has size m by n and rank m. f is not necessarily differentiable or convex. Show that the problem can be formulated as an equivalent problem with m variables, by making change of y = Ax. 
January 26th, 2017, 05:30 AM  #2 
Senior Member Joined: Apr 2014 From: Glasgow Posts: 2,090 Thanks: 701 Math Focus: Physics, mathematical modelling, numerical and computational solutions 
Since $\displaystyle y = Ax, x = A^{1} y$, $\displaystyle A^{1} A = 1$ and $\displaystyle (AB)^T = B^T A^T$ $\displaystyle f(Ax) + x^T x$ $\displaystyle = f(y) + (A^{1} y)^T A^{1} y$ $\displaystyle = f(y) + y^T (A^{1})^T A^{1} y$ This is almost exactly the same as the original equation except that y is the new variable of interest and you have to work out $\displaystyle (A^{1})^T A^{1}$ as the filling of the matrix sandwich Last edited by Benit13; January 26th, 2017 at 05:41 AM. 

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